THE PYRAMID DECOMPOSITION AND RANK MINIMIZATION

It is shown that every m X n matrix M over a regular ring with 1 can b e decomposed as M = M1 + M2 + M3 + M4, where M1 less-than-or-equal-to M1 + M2 less-than-or-equal-to M1 + M2 + M3 less-than-or-equal-to M and less-than-or-equal-to is the minus order. This horizontal pyramid dec omposition can be used to obtain, firstly, a pyramid decomposition for 2 x 2 block matrices, which over a division ring immediately yields t he classical rank formula for block matrices. Secondly, it also yields a horizontal version of this block rank formula. These rank formulae are then respectively used to solve the horizontal and vertical rank m inimization problems, which are involved in the computation of the sho rted matrix of M relative to subspaces W1 and W2.